Positive Polynomial Ideals

• Positive polynomial ideals are defined either by nonnegativity on squares in graded rings or by order preservation in Banach-lattice settings, forming the basis for distinct sums-of-squares and optimization theories.
• They underpin composition theorems and factorization results, providing tight rank bounds and structural insights applicable to truncated moment problems and real Waring decompositions.
• These frameworks highlight varying notions of positivity that fuel research in real algebraic geometry, nonlinear operator theory, and the development of positive polynomial methodologies.

The phrase positive polynomial ideals appears in distinct technical settings. In Banach-lattice theory it denotes classes of positive $m$-homogeneous polynomials between Banach lattices that are stable under composition by positive or regular operators (Bounabab et al., 4 Sep 2025). A closely related real-algebraic notion is the positive Gorenstein ideal: a Gorenstein ideal in the graded ring $\mathbb{R}[x_1,\dots,x_n]$ with socle in degree $2d$ whose socle functional is nonnegative on squares (Blekherman, 2012). In the first setting, the theory develops basic principles, composition theorems, and constructions from positive operator ideals; in the second, positive Gorenstein ideals arise naturally in the context of nonnegative polynomials and sums of squares and are applied to real algebraic geometry, analysis, and optimization (Blekherman, 2012, Bounabab et al., 4 Sep 2025).

1. Two principal frameworks

The two principal frameworks differ in ambient category, but each places positivity on a structure canonically attached to polynomials.

Framework	Ambient setting	Positivity condition
Positive Gorenstein ideals	$\mathbb{R}[x_1,\dots,x_n]$, graded commutative algebra	The socle functional $\ell$ satisfies $\ell(p^2)\ge 0$ for every $p\in \mathbb{R}[x]_d$
Positive $m$-homogeneous polynomial ideals	Banach lattices $E,F$ and Banach spaces $X,Y$	$\mathbb{R}[x_1,\dots,x_n]$0, with ideal properties under positive composition

In the graded-algebraic setting, positivity is imposed on the socle functional of an Artinian Gorenstein quotient. In the Banach-lattice setting, positivity is imposed directly on the values of a polynomial and on the admissible operator compositions defining the ideal structure. This distinction is decisive: the former is tied to apolarity, sums of squares, truncated moments, and real Waring decompositions, whereas the latter extends positive operator-ideal theory to the nonlinear setting (Blekherman, 2012, Bounabab et al., 4 Sep 2025).

A useful consequence of juxtaposing these frameworks is terminological clarity. The same adjective positive governs either nonnegativity on squares or order preservation on positive cones, not a single universal notion of positivity across all polynomial-ideal theories. This suggests that the subject is best read as a family of related positivity theories rather than a single unified definition.

2. Positive Gorenstein ideals and apolarity

Let $\mathbb{R}[x_1,\dots,x_n]$1 be the standard $\mathbb{R}[x_1,\dots,x_n]$2-graded polynomial ring, and fix $\mathbb{R}[x_1,\dots,x_n]$3. A Gorenstein ideal $\mathbb{R}[x_1,\dots,x_n]$4 with socle in degree $\mathbb{R}[x_1,\dots,x_n]$5 is an ideal such that the quotient algebra $\mathbb{R}[x_1,\dots,x_n]$6 has Hilbert function

$\mathbb{R}[x_1,\dots,x_n]$7

and top graded piece $\mathbb{R}[x_1,\dots,x_n]$8 one-dimensional. Equivalently, the natural pairing

$\mathbb{R}[x_1,\dots,x_n]$9

descends to a perfect duality between $2d$0 and $2d$1. The generator of $2d$2 defines a linear functional

$2d$3

called the socle of $2d$4, and one writes $2d$5 when the apolar kernel is emphasized (Blekherman, 2012).

A positive Gorenstein ideal is then a Gorenstein ideal $2d$6 with socle $2d$7 such that

$2d$8

Equivalently, the quadratic form

$2d$9

is positive semidefinite. The nonnegativity-on-squares condition can also be written as

$\mathbb{R}[x_1,\dots,x_n]$0

where $\mathbb{R}[x_1,\dots,x_n]$1 (Blekherman, 2012).

The apolar viewpoint identifies $\mathbb{R}[x_1,\dots,x_n]$2 with its algebraic dual by sending a form

$\mathbb{R}[x_1,\dots,x_n]$3

to the differential operator

$\mathbb{R}[x_1,\dots,x_n]$4

The apolar ideal is

$\mathbb{R}[x_1,\dots,x_n]$5

and it is Gorenstein with socle $\mathbb{R}[x_1,\dots,x_n]$6. Since

$\mathbb{R}[x_1,\dots,x_n]$7

a Gorenstein ideal is positive precisely when it is of the form $\mathbb{R}[x_1,\dots,x_n]$8 and the differential operator $\mathbb{R}[x_1,\dots,x_n]$9 is nonnegative on squares. This exact equivalence connects positivity in the quotient algebra to positivity of a differential operator on $\ell$0 (Blekherman, 2012).

3. Extremal geometry, rank bounds, and model examples

The structural theory of positive Gorenstein ideals is organized around maximal positive Gorenstein ideals, meaning those whose socle spans an extreme ray of the dual cone of sums of squares in degree $\ell$1. If $\ell$2 is maximal positive Gorenstein with socle $\ell$3 of degree $\ell$4, and $\ell$5 is not a point-evaluation, then the forms in $\ell$6 have no common zeroes, real or complex, and they generate $\ell$7. Moreover, there is a sharp lower bound

$\ell$8

In dual-cone language, if

$\ell$9

then any $\ell(p^2)\ge 0$0 for which $\ell(p^2)\ge 0$1 has rank strictly less than $\ell(p^2)\ge 0$2 for $\ell(p^2)\ge 0$3, or less than $\ell(p^2)\ge 0$4 for $\ell(p^2)\ge 0$5, must be a point-evaluation. These bounds are tight (Blekherman, 2012).

The same geometry yields rank thresholds in two standard problems. For the truncated moment problem, if $\ell(p^2)\ge 0$6 satisfies $\ell(p^2)\ge 0$7 and

$\ell(p^2)\ge 0$8

for $\ell(p^2)\ge 0$9, or $p\in \mathbb{R}[x]_d$0 for $p\in \mathbb{R}[x]_d$1, then $p\in \mathbb{R}[x]_d$2 arises from integration against a positive measure supported on finitely many points, in fact exactly $p\in \mathbb{R}[x]_d$3 real points. For real Waring decomposition, if $p\in \mathbb{R}[x]_d$4 has middle-catalecticant $p\in \mathbb{R}[x]_d$5 of rank less than $p\in \mathbb{R}[x]_d$6, or less than $p\in \mathbb{R}[x]_d$7 when $p\in \mathbb{R}[x]_d$8, then

$p\in \mathbb{R}[x]_d$9

with $m$0, and the bound on Waring rank is sharp (Blekherman, 2012).

The tightness of the codimension estimates is exhibited by explicit low-degree constructions. In the ternary case $m$1 with $m$2, if $m$3 is a smooth cubic and $m$4 is a general form of degree $m$5 such that $m$6 is a transverse intersection of $m$7 real points in $m$8, then the corresponding extreme functional $m$9 has apolar ideal $E,F$0 with

$E,F$1

In the quartic case $E,F$2, $E,F$3, a complete intersection of four real quadrics in $E,F$4 cuts out $E,F$5 real points in $E,F$6, and the corresponding Gorenstein ideal has socle in degree $E,F$7 with $E,F$8 of codimension

$E,F$9

These examples realize the boundary cases of the general theory (Blekherman, 2012).

4. Sums of squares, Hilbert’s theorem, and optimization

Positive Gorenstein ideals were introduced precisely because they provide a framework for studying concrete aspects of sums-of-squares representations. One of the main applications is a simple proof of Hilbert’s nearly forgotten result on representations of ternary nonnegative forms as sums of squares of rational functions. In the notation of cones of nonnegative forms and sums of squares, the result states that for every $X,Y$0 there exists $X,Y$1 such that

$X,Y$2

The proof strategy proceeds by separation: if no such $X,Y$3 existed, one would separate $X,Y$4 from the linear subspace $X,Y$5 by an extreme positive functional $X,Y$6 of degree $X,Y$7; the associated positive Gorenstein ideal $X,Y$8 then cannot contain the strictly positive form $X,Y$9, giving a contradiction (Blekherman, 2012).

The same rank bounds furnish a stopping criterion in polynomial optimization. In Lasserre’s hierarchy of sum-of-squares relaxations, each level $\mathbb{R}[x_1,\dots,x_n]$00 produces an optimal dual functional $\mathbb{R}[x_1,\dots,x_n]$01. If the associated moment matrix has rank at most $\mathbb{R}[x_1,\dots,x_n]$02, or at most $\mathbb{R}[x_1,\dots,x_n]$03 when $\mathbb{R}[x_1,\dots,x_n]$04, then $\mathbb{R}[x_1,\dots,x_n]$05 is a sum of point-evaluations, and the relaxation is exact. In this regime, no higher-degree relaxation is needed (Blekherman, 2012).

The Waring-rank application has a similarly certificate-like form. If the middle catalecticant of a form $\mathbb{R}[x_1,\dots,x_n]$06 is positive semidefinite of sufficiently small rank, then the theory produces the exact real Waring decomposition by $\mathbb{R}[x_1,\dots,x_n]$07-th powers of real linear forms. This places positive Gorenstein ideals at the interface of convex geometry, apolarity, and explicit decomposition theory (Blekherman, 2012).

5. Positive $\mathbb{R}[x_1,\dots,x_n]$08-homogeneous polynomial ideals on Banach lattices

In Banach-lattice theory, the basic object is an $\mathbb{R}[x_1,\dots,x_n]$09-homogeneous polynomial

$\mathbb{R}[x_1,\dots,x_n]$10

between Banach lattices $\mathbb{R}[x_1,\dots,x_n]$11 and $\mathbb{R}[x_1,\dots,x_n]$12, characterized by the existence of a unique symmetric $\mathbb{R}[x_1,\dots,x_n]$13-linear map

$\mathbb{R}[x_1,\dots,x_n]$14

such that $\mathbb{R}[x_1,\dots,x_n]$15. Equivalently,

$\mathbb{R}[x_1,\dots,x_n]$16

The norm is

$\mathbb{R}[x_1,\dots,x_n]$17

and $\mathbb{R}[x_1,\dots,x_n]$18 is positive when

$\mathbb{R}[x_1,\dots,x_n]$19

This is the basic order-theoretic positivity notion in the nonlinear setting (Bounabab et al., 4 Sep 2025).

A polynomial ideal $\mathbb{R}[x_1,\dots,x_n]$20 assigns to each pair $\mathbb{R}[x_1,\dots,x_n]$21 a subspace $\mathbb{R}[x_1,\dots,x_n]$22 containing all finite-type polynomials and satisfying the ideal property

$\mathbb{R}[x_1,\dots,x_n]$23

with a compatible ideal norm. A positive polynomial ideal is obtained by restricting one or both of the compositional operators $\mathbb{R}[x_1,\dots,x_n]$24 to positive or regular operators. Thus a positive left polynomial ideal $\mathbb{R}[x_1,\dots,x_n]$25 requires $\mathbb{R}[x_1,\dots,x_n]$26; a positive right polynomial ideal $\mathbb{R}[x_1,\dots,x_n]$27 requires $\mathbb{R}[x_1,\dots,x_n]$28; and a positive two-sided ideal $\mathbb{R}[x_1,\dots,x_n]$29 requires both $\mathbb{R}[x_1,\dots,x_n]$30 and $\mathbb{R}[x_1,\dots,x_n]$31 (Bounabab et al., 4 Sep 2025).

The basic closure result is that if $\mathbb{R}[x_1,\dots,x_n]$32 is a positive left polynomial ideal and $\mathbb{R}[x_1,\dots,x_n]$33 is a positive right polynomial ideal, then the composition class

$\mathbb{R}[x_1,\dots,x_n]$34

is again a positive polynomial ideal, with norm

$\mathbb{R}[x_1,\dots,x_n]$35

This composition norm satisfies linearity, scaling, the ideal inequality

$\mathbb{R}[x_1,\dots,x_n]$36

for positive $\mathbb{R}[x_1,\dots,x_n]$37, and the dominance estimate $\mathbb{R}[x_1,\dots,x_n]$38 (Bounabab et al., 4 Sep 2025).

A central construction starts from a positive operator ideal $\mathbb{R}[x_1,\dots,x_n]$39. One defines

$\mathbb{R}[x_1,\dots,x_n]$40

with norm

$\mathbb{R}[x_1,\dots,x_n]$41

If $\mathbb{R}[x_1,\dots,x_n]$42 is a positive right Banach ideal, then $\mathbb{R}[x_1,\dots,x_n]$43 is a positive right Banach polynomial ideal. Dually, if $\mathbb{R}[x_1,\dots,x_n]$44 is a positive left operator ideal, one may define $\mathbb{R}[x_1,\dots,x_n]$45 by pre-composition (Bounabab et al., 4 Sep 2025).

6. Domination classes, factorization theorems, and linearization

Several concrete positive polynomial ideals are obtained by combining domination inequalities with factorization. A polynomial $\mathbb{R}[x_1,\dots,x_n]$46 is Cohen positive strongly $\mathbb{R}[x_1,\dots,x_n]$47-summing if there is $\mathbb{R}[x_1,\dots,x_n]$48 such that for every finite choice $\mathbb{R}[x_1,\dots,x_n]$49, $\mathbb{R}[x_1,\dots,x_n]$50,

$\mathbb{R}[x_1,\dots,x_n]$51

Its norm is $\mathbb{R}[x_1,\dots,x_n]$52, and the class satisfies

$\mathbb{R}[x_1,\dots,x_n]$53

where $\mathbb{R}[x_1,\dots,x_n]$54 is the positive $\mathbb{R}[x_1,\dots,x_n]$55-summing left operator ideal (Bounabab et al., 4 Sep 2025).

Positive Cohen $\mathbb{R}[x_1,\dots,x_n]$56-nuclear polynomials form another factorization class. Denoting this space by $\mathbb{R}[x_1,\dots,x_n]$57, one has

$\mathbb{R}[x_1,\dots,x_n]$58

so these polynomials factor through a positive $\mathbb{R}[x_1,\dots,x_n]$59-summing operator. Likewise, a positive $\mathbb{R}[x_1,\dots,x_n]$60-dominated polynomial class $\mathbb{R}[x_1,\dots,x_n]$61 is characterized by

$\mathbb{R}[x_1,\dots,x_n]$62

meaning every positive $\mathbb{R}[x_1,\dots,x_n]$63-dominated polynomial factors as $\mathbb{R}[x_1,\dots,x_n]$64 with $\mathbb{R}[x_1,\dots,x_n]$65, $\mathbb{R}[x_1,\dots,x_n]$66, and

$\mathbb{R}[x_1,\dots,x_n]$67

These identities are explicitly described as polynomial analogues of operator-ideal factorizations (Bounabab et al., 4 Sep 2025).

For positive $\mathbb{R}[x_1,\dots,x_n]$68-dominated polynomials, denoted $\mathbb{R}[x_1,\dots,x_n]$69, the theory provides both a Pietsch-domination theorem and a Kwapień-type factorization. The domination theorem states that there exist probability measures $\mathbb{R}[x_1,\dots,x_n]$70 on $\mathbb{R}[x_1,\dots,x_n]$71 and $\mathbb{R}[x_1,\dots,x_n]$72 on $\mathbb{R}[x_1,\dots,x_n]$73 yielding the corresponding two-measure estimate for $\mathbb{R}[x_1,\dots,x_n]$74, and the best constant is the ideal norm $\mathbb{R}[x_1,\dots,x_n]$75. The factorization theorem is

$\mathbb{R}[x_1,\dots,x_n]$76

with

$\mathbb{R}[x_1,\dots,x_n]$77

At the structural level, Proposition 2.9 gives a linearization characterization: $\mathbb{R}[x_1,\dots,x_n]$78 if and only if its linearization

$\mathbb{R}[x_1,\dots,x_n]$79

lies in $\mathbb{R}[x_1,\dots,x_n]$80, and in particular

$\mathbb{R}[x_1,\dots,x_n]$81

isometrically (Bounabab et al., 4 Sep 2025).

These results place positive polynomial ideals among the nonlinear analogues of positive linear and multilinear operator ideals. The stated applications include analysis of nonlinear mappings between Banach lattices preserving order-structure, extensions of Banach-lattice operator theory to polynomial and holomorphic cases, factorization and summability results for entire functions on lattices, and further development of interpolation, duality, and tensor-product techniques in the positive nonlinear regime (Bounabab et al., 4 Sep 2025).

7. Terminological boundaries and open directions

A recurrent source of ambiguity is that positive may refer either to positivity in the order or nonnegativity sense, or merely to characteristic $\mathbb{R}[x_1,\dots,x_n]$82. The work on ideals preserved by linear changes of coordinates in positive characteristic studies ideals in a polynomial ring over an algebraically closed field of characteristic $\mathbb{R}[x_1,\dots,x_n]$83, classified by carry patterns introduced by Doty; these are GL$\mathbb{R}[x_1,\dots,x_n]$84-invariant ideals generated from degree-$\mathbb{R}[x_1,\dots,x_n]$85 monomials whose carry patterns lie below a specified element of the finite lattice $\mathbb{R}[x_1,\dots,x_n]$86 (Cattell-Ravdal et al., 2024). This is a different theory from positive Gorenstein ideals and positive $\mathbb{R}[x_1,\dots,x_n]$87-homogeneous polynomial ideals: here the adjective positive refers to the characteristic of the ground field, not to nonnegativity on squares or order-preserving polynomial maps.

Within the real-algebraic theory, several open directions are explicit. Sharp codimension bounds are known for socle in degree $\mathbb{R}[x_1,\dots,x_n]$88 and in the cases stated for larger $\mathbb{R}[x_1,\dots,x_n]$89, but understanding minimal Hilbert functions of positive Gorenstein ideals in higher socle degrees remains open. The current tight examples come from full real transverse intersections, suggesting further study of nontransverse intersections and possible new extremal rays of $\mathbb{R}[x_1,\dots,x_n]$90. Exact degree bounds for multipliers are settled for ternary forms via Hilbert’s bound $\mathbb{R}[x_1,\dots,x_n]$91, but for $\mathbb{R}[x_1,\dots,x_n]$92, $\mathbb{R}[x_1,\dots,x_n]$93 it is still unknown whether quadratic multipliers always suffice. The same perspective also points to possible connections with K3 surfaces, as well as noncommutative and matrix-valued generalizations in free $\mathbb{R}[x_1,\dots,x_n]$94-algebras and operator theory (Blekherman, 2012).

Within the Banach-lattice theory, the current emphasis is foundational: definitions, closure properties, factorization constructions from operator ideals, and canonical examples. This suggests continued development of interpolation, duality, and tensor-product methods, and broader extensions from polynomial classes to holomorphic mappings on lattices (Bounabab et al., 4 Sep 2025).

Taken together, these developments show that positive polynomial ideals is not a single notion but a family of rigorously formulated positivity structures on polynomial objects. In one line, positivity is encoded by a socle functional nonnegative on squares and exploited through apolarity and convex geometry; in another, it is encoded by order preservation and positive composition in Banach lattices and developed through factorization and domination theory.